2026/05/13 by Alexander Pigazzini, Magdalena Toda
#math.DG
We show that the cohomological invariant r^\sharp, introduced in [1] as a lower bound for the off-diagonal holonomy dimension of metric connections with totally skew torsion on product manifolds, predicts the behaviour of the torsion 3-form under both dimensional reduction and Buscher T-duality. On a product M = Σg × M2 equipped with a product metric, when r^\sharp = 0 the parallel-form strata identify a flat circle factor S1β⊂ M2 via the de Rham splitting theorem, and the entire H-flux is converted into geometric flux under T-duality along S1β (the parallel regime); when r^\sharp = 1, no such circle factor exists, and the H-flux survives T-duality along every flat circle factor as H-flux in the dual background (the transversely non-reducible regime). When M2 = N × Tk contains a torus factor, we prove that the Bouwknegt--Evslin--Mathai obstruction to successive T-dualities vanishes automatically for H-flux of pure bidegree (2,1), that the resulting dualities are non-interfering and order-independent, and that r^\sharp detects the irreducible kernel of the H-flux: the component that survives T-duality along every flat circle factor and cannot be converted into geometric or non-geometric flux in any duality frame. This provides a metric refinement of topological T-duality: while the latter disregards the Riemannian metric entirely, r^\sharp detects whether the cohomological coupling is aligned with the flat sub-factors identified by the Levi-Civita parallel-form strata.